Tutorial 12 - May 30, 2014
Questions related to SECTION 8.1
1. (a) Find the next two terms of the sequence.
(b) Find a recurrence relation that generates the relation.
(c) Find an explicit formula for the general nth term of the sequence.
i.
1 1 1 1
1, ,
MATH151 Final Exam
Department of Mathematics
Spring 2014 - June, 9, 2014
1. Evaluate the area between the curves:
(a)
y = x3
x2
2x;
y = 0;
1
x
2
Solution. The points of intersection of two graphs are found as
x3
x2
2x = 0;
x = 0;
x x2 x
x = 1;
2 = x (x
x
Examples for Graph Sketching
Mustafa Riza
October 18, 2012
1
Graph of a Polynomial
Sketch the graph of y = x4 12x3 + 48x2 64x = x(x 4)3 .
1. Domain: x R
2. x- and y-intercepts:
x-intercept:
y=0
x(x 4)3 = 0 x = 0 or x = 4.
So the x-intercepts are: (0, 0)
g .q
e
qs .r
y .ZZ_6I
suollcun!
:
1e1 'e;n8g eqi ur u.tr.oqi sr
/;o
qder8 eq1 suogiong eary
'7,r
Questions related to SECTION 5.3
:t
Tutorial 8 - May 2, 2014
suortrunJeeruo^Alaqry(t)J,r[
= (r).cfw_ puerye)!,0[
:
(r)V
1. The graph of f is shown in the gure
Tutorial 10 - May 16, 2014
Questions related to SECTION 7.3
1. Evaluate the following integrals.
(a)
1
dx
1 x2
Let
x = sin
Note that :
dx = cos d
1 x2 = 1 sin2 = cos2 = cos
Thus we have
1
dx =
1 x2
Since we know that x = sin
1
cos d =
cos
d = + C
= s
Tutorial 9 - May 9, 2014
Questions related to SECTION 6.2
1. Determine the area of the shaded regions.
(a)
First of all let us nd
3
2
1
0
1
2
3
the intersection point of the two curves, i.e.
2x = 3 x
x=1
Thus we have
1
x
(3x2 )dx =
0
x2
2x
3x
2
ln 2
1
=
D10
APPENDIX D
SECTION
D.2
Precalculus Review
The Cartesian Plane
The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles
The Cartesian Plane
An ordered pair x, y of real numbers has x as its first member and y as its second
member. Th
APPENDIX E
Rotation and the General
Second-Degree Equation
Rotation of Axes Invariants Under Rotation
y
y
Rotation of Axes
In Section 9.1, you learned that equations of conics with axes parallel to one of the
coordinate axes can be written in the general
EASTERN MEDITERRANEAN UNIVERSITY
FACULTY OF ARTS AND SCIENCES DEPARTMENT OF MATHEMATICS
2014-2015 SPRING SEMESTER
COURSE CODE
MATH152
COURSE TITLE
Calculus-II
COURSE TYPE
University Core (UC)
LECTURERS
Groups 1 & 5
Groups 2 & 8
Group 3
Group 6
Group 7
ASS
APPENDIX F
Complex Numbers
Operations with Complex Numbers Complex Solutions of Quadratic Equations
Polar Form of a Complex Number Powers and Roots of Complex Numbers
Operations with Complex Numbers
Some equations have no real solutions. For instance, th
Tutorial 11 - May 23, 2014
Questions related to SECTION 7.7
1. Evaluate the following integrals or state that they diverge.
x2 dx
(a)
1
b
x
2
dx = lim
b
1
(b)
0
x
2
1
x
dx = lim
b
1
b
b
1
0
b
dx
= lim
(x + 1)3 b
0
b
2
=
0
1
1
+
2
2(b + 1)
2
=
1
2
dx
x ln
MATH151 Quiz 4
Department of Mathematics, Spring 2014
May 26, 2014 Duration 30 min
Questions:
1. Evaluate using integration by parts
Z
u = x; dv = cos xdx
x cos xdx =
du = dx; v = sin x
= x sin x + cos x + C
= x sin x
Z
sin xdx
2. Evaluate using partial f
Tutorial 6 - April 04, 2014
Questions related to SECTION 4.3
1. Sketch the graph of f (x) = x4 6x2
(a) Domain: (, )
(b) x and y intercepts:
xintercept: y = 0
x4 6x2 = 0
So the xintercepts are (0, 0),
yintercept: x = 0
( 6, 0),
x = 0,
x = 6,
x=
6
( 6, 0)
y
Guidelines for Curve Sketching
1. Find the domain of f (x), i.e. all real numbers where f (x) is dened.
2. Find x- and y- intercepts
x-intercept(s) are the solutions of the equation f (x) = 0, if any
y-intercept is the value of function f (0), if it exi
Tutorial 1 - February 28, 2014
Questions related to SECTION 2.2
1. Let
f (x) =
x3 1
x1
(a) Calculate f (x) for each value of x in the following table.
x
0.9 0.99
f(x) 2.7 2.97
0.999 0.9999
2.997 2.9997
1
1.0001 1.001
undened 3.0003 3.003
1.01
1.1
3.0301 3
Tutorial 2 - March 7, 2014
Questions related to SECTION 2.4
1. The graph of g in the gure has vertical asymptotes at x = 2 and x = 4. Find the
following limits if possible.
y
1
2
3
4
5
6
x
(a) lim g(x) =
x2
(b) lim g(x) =
+
x2
(c) lim g(x)
x2
does
not
e
Tutorial 3 - March 14, 2014
Questions related to SECTION 2.6
1. Determine whether the following functions are continuous at a.
(a) f (x) =
x2 1
,
x1
3,
if x = 1
if x = 1
a=1
Any function f (x) is continuous at a given point a iff
lim f (x) = f (a)
xa
Thus
Tutorial 5 - March 28, 2014
Questions related to SECTION 4.1
For the function
f (x) =
(x2
x
+ 1)2
on [2, 2]
(a) Find the critical points of f on the given interval.
(b) Determine the absolute extreme values of f on the given interval.
(c) Use a graphing u
MATH151 Quiz 3
Department of Mathematics, Spring 2014
April 12, 2014
Questions:
1. Use linear approximation of the function f (x) =
Choose a value of a to produce a small error.
a
=
f (27)
=
=
p
3
x to estimate
p
3
1
1
f 0 (x) = p ; f 0 (27) =
;
3
2
27
3
Tutorial 7 - April 25, 2014
Questions related to SECTION 4.7
1. Evaluate the following limits.
3 sin 4x
x0
5x
(a) lim
3 sin 4x
=
x0
5x
0
0
12 cos 4x
12
= lim
=
x0
5
5
lim
sin x
= 1.
x0
x
Recall also that lim
Thus we can evaluate the limit as:
3 sin 4x
3
s
9.1:Approximating Functions with Polynomials
1. a) Find the linear approximating polynomial for the following function centered at the
given point a
b) Find the quadratic approximatin polynomial for the following function centered at
the given point a
c)